Download Asymptotic Representation of Relaxation Oscillations in by Elena V. Grigorieva, Sergey A. Kaschenko PDF

By Elena V. Grigorieva, Sergey A. Kaschenko

In this booklet we research leisure oscillations in types of lasers with nonlinear components controlling mild dynamics. The types are in keeping with fee equations taking into consideration periodic modulation of parameters, optoelectronic not on time suggestions, mutual coupling among lasers, intermodal interplay and different factors.

With the purpose to review leisure oscillations we current the targeted asymptotic approach to integration for usual differential equations and differential-difference equations. therefore, they're diminished to discrete maps. studying the maps we describe analytically such nonlinear phenomena in lasers as multistability of large-amplitude rest cycles, bifurcations of cycles, managed switching of regimes, part synchronization in an ensemble of coupled structures and others.

The ebook might be fruitful for college students and technicians in nonlinear laser dynamics and in differential equations.

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14) where h ∈ (0, qk ] and c ∈ (1 + h, q]. Note, parameter c should exceed (1 + h) by a finite value which does not depend on v, thus u (0) > 0. 1. To simplify the following expressions, all solutions u(t, c, h), y(t, c, h), k(t, c, h) with initial conditions from set S(c, h), will be hereupon denoted as u(t), y(t), k(t), respectively. Let us also denote t1 , t2 ... successive positive moments of time, when u(ti ) = 1, i = 0, 1, 2 . .. 1. Within the [0, t1 ]-interval the function u(t) takes asymptotically large values, u(t) 1 all over, except for the small vicinity of the interval ends.

That allowed to observe lasing on an unstable orbit during tens oscillation periods. In terms of mapping dynamics, the switching was conditioned by shifting the phase point to a position on a stable separatrix. 45) is shown in Fig. 12. In this figure attractors (stable fixed points and cycles) are marked with black circles, while the unstable points (unstable cycles) are marked with white circles. 04, ω = 30. The attractors are marked with black circles, while the unstable points (unstable cycles) are marked with white circles.

46) Let us designate the coordinates of such fixed points as (cn , ϕn ). 47) where the pulse energy takes the form pn = ln[cn /(cn − (q − 1)Tn )]. Since ϕn are defined through cos ϕn , then to each cn correspond two phase values + − ϕn = {ϕ− n < π, ϕn = 2π − ϕn }. In laser experiments the characteristic modulation frequency is such, that modulation period may be regarded as quite small value, therefore Tn 1 is also small for small n. It is useful to expand the obtained expressions in a Spiking in Single-Mode Laser 50 power series of Tn , cn = cos ϕn = 1 1 + (q − 1)Tn + O(Tn2 ), 2 1 (q − 1)q 2 2 Tn + O(Tn3 ) .

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